By transforming a picture or sound signal, samples of the signal are transformed into coefficients. The coefficients are transmitted and subsequently backward-transformed to reconstruct the samples of the picture or sound signal. Both the forward transform at the transmitter end and the backward transform at the receiver end are defined by a transform matrix.
In a non-lapped transform such as the Discrete Cosine Transform (DCT) which is widely used in image coding, series of L samples are transformed into the same number of coefficients. From each series of coefficients, the corresponding series of samples can be reconstructed by a backward transform at the receiving end. The forward and backward transform matrices each have a dimension of L*L elements.
In a lapped transform, overlapping series of samples of the signal are transformed into a number of coefficients less than the number of samples. For example, a series of L samples partly overlapping each other are transformed into M coefficients. The transform matrix at the transmitter end and at the receiver end each have a dimension of M*L elements.
A signal transform may also be considered as a special case of filter bank coding. For example, the above-mentioned lapped transform may be implemented by a filter bank comprising M filters each having a filter length L. The filtered signals are decimated by a factor M which means that only each Mth sample of the filtered signals constitutes a coefficient which is transmitted while the intermediate M-1 samples are ignored. At the receiver end, the M coefficients are up-sampled (filling in the M-1 intermediate samples of with the value zero) and then applied to M interpolation filters each having a filter length L. The interpolated signals are subsequently summed in an adder.
By a transform, the input signal is decomposed into a weighted sum of basis functions, the coefficients constituting the weighting factors. The rows of the transform matrix or, equivalently, the impulse response functions of the interpolation filters of the filter bank constitute the basis functions. Each basis function has a frequency spectrum. Generally, the basis functions are chosen to be such that each basis function comprises a part of the total frequency spectrum.
Currently, segmentation based coding systems are envisaged in which different segments (e.g. subpictures of an image) are subjected to different transforms, for example in response to characteristics of a picture signal. One segment of the signal may be subjected to a first transform whereas a different second transform may be desired for a subsequent segment of the signal. A different transform is not only understood to mean a different transform matrix size but also a same matrix size with different element values. Switching from one transform to another does not cause any problem in a transmission system employing a non-lapped transform, because each series of samples can be reconstructed from the same number of coefficients, provided that the actual transform is known at the receiver end. However, problems do arise in transmission systems employing the lapped transform. Signal samples at the transition can not be reconstructed from the coefficients obtained by different transforms without complications. Similar problems arise at the boundary of signals having a finite extent, such as picture signals.